On wikipedia's 'Arity' page I noticed the statement,
The arity of a relation (or predicate) is the dimension of the domain in the corresponding Cartesian product.
I do not understand what this means and do not see an explanation on that page.
What is a relation's corresponding Cartesian product?
What is the dimension of this Cartesian product's domain?
On
By definition:
an $n$-ary relation on $n$ sets, is any subset of Cartesian product of the $n$ sets.
so, as an example: $$ R_1(x,y,z)=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+z^2\ge 1\} $$
is a relation of arity=$3$
It is the number of inputs to the relation or predicate. The term is also used with functions, while I have not seen relations with other than two inputs. The relation "less than" needs two values to make a complete sentence such as $x \lt y$, so has arity two. The function $\sin$ takes one input to make a value like $\sin x$ so has arity one. The distance function in $\Bbb R^3, \sqrt{x^2+y^2+z^2}$ takes three inputs, so has arity three.